Classification the elements of the twisted Hessian curves in the ring Fq[e], e³=e²

Let Fq denote the finite field of q elements, where q is a prime power. In this paper, we study the twisted Hessian curves denoted Ha,d(Fq[e]) over the ring Fq[e], where e³=e² and (a,d)∈ (Fq[e])². More precisely, we study some arithmetical properties of this ring and using the Twisted Hessian equation, we define the twisted Hessian curves Ha,d(Fq[e]). This work study the twisted Hessian curve helped us to define two twisted Hessian over the finite field Fq. We end this paper by giving the classification of the elements in Ha,d(Fq[e]).